The characteristic function in (REF ) resembles the characteristic function of the AL univariate distribution discussed in [1]} and [2]} where \(\mu ^\star , \tau ^\star , \delta ^\star \)  are the scale, skewness and scale parameters, respectively. Therefore, the characteristic function of \(Y^{b}\)  in (REF ) can be rewritten as the characteristic function of a univariate AL distribution with parameters:
\(\mu ^\star = b^{\prime }\mu  \mbox{,} \quad \tau ^\star = \frac{1}{2}\left(1-\frac{b^{\prime }D\tilde{\xi }}{\sqrt{2(b^{\prime }D\tilde{\Sigma }Db) + (b^{\prime }D\tilde{\xi })^2}}\right) \quad \mbox{and} \quad \delta ^\star =\frac{(b^{\prime }D\tilde{\Sigma }Db)}{2{\sqrt{2(b^{\prime }D\tilde{\Sigma }Db) + (b^{\prime }D\tilde{\xi })^2}}}. \nonumber \\[-0.01in]\) 
